#7052088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#45136	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1494 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 48 points:

(2,6)(7,8)(9,10,14,11)(12,15,13,16)(17,18)(19,27)(20,31)(21,25)(22,26)(23,30)(24,28)(29,42)(32,41)(33,47)(34,39)(35,38)(36,48)(37,44)(40,45)(43,46);(1,2)(3,7)(4,8)(5,6)(10,11)(12,13)(17,19,22,30)(18,23,26,27)(20,32,34,28)(21,35,37,29)(24,39,41,31)(25,42,44,38)(33,45,36,46)(40,47,43,48);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,15,13,16)(17,20,19,28,22,34,30,32)(18,24,23,31,26,41,27,39)(21,36,29,46,37,33,35,45)(25,43,38,48,44,40,42,47);(1,3,5,4)(2,8,6,7)(9,12,14,13)(10,15,11,16)(17,21,22,37)(18,25,26,44)(19,29,30,35)(20,33,34,36)(23,38,27,42)(24,40,41,43)(28,45,32,46)(31,47,39,48);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,15,11,16)(18,26)(20,34)(21,37)(23,27)(28,32)(29,35)(40,43)(47,48)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=fgi,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6959284);